The local linear approximation to the function $f$ at $x=-2$ is $y=3x-5$. What is the value of $f(-2)+f'(-2)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $-7$ (Choice B) B $-8$ (Choice C) C $-9$ (Choice D) D $-10$
The local linear approximation of $f$ at $x=-2$ is achieved using the equation of the line tangent to $f$ at $x=-2$. In other words, $y=3x-5$ is the equation of the line tangent to the graph of $f$ at $x=-2$. How can we use this to find $f(-2)$ and $f'(-2)$ ? Since the line is tangent to the graph of $f$ at $x=-2$, we know two key facts about it: The line passes through the point $({-2},{f(-2)})$ The line's slope is ${f'(-2)}$ The slope of $y={3}x-5$ is ${3}$. The $y$ -value that corresponds to $x={-2}$ is $3({-2})-5={-11}$. Now we can find our answer: ${f(-2)}+{f'(-2)}={-11}+{3}=-8$